El Naschie’s coherence on the subquantum medium

In the hydrodynamic formulation of the Scale Relativity theory one shows that a stable vortices distribution of bipolaron type induces superconducting pairs by means of the quantum potential. One builds the superconducting fractal by an iterated map and demonstrates that the superconducting pairs results as projections of this fractal. Thus, usual mechanisms (as example the exchange interaction used in the bipolaron theory) are reduced to the coherence on the subquantum medium in a ε^(∞) space (El Naschie’s coherence).     

Quantum decoherence and El Naschie’s complex temporality

It is well-known that the Schrödinger equation reduces to a classical diffusion equation by means of Wick rotation (t → it), suggesting a correspondence between quantum and classical mechanics. Nonetheless, this result does not admit a clear conceptual interpretation. In the framework of his fractal space-time theory, El Naschie showed that great conceptual advantage could be achieved by extending the imaginary time, it, to a perfectly symmetric, complex conjugate time 0 ± it. In this note we show through a simple analysis, involving formal analytic continuation (t → 0 ± it), that El Naschie’s time complexification provides the basis for a physical interpretation of the correspondence between quantum and classical mechanics in terms of quantum decoherence. We find that decoherent states inevitably arise due to time symmetry breaking as we go from the micro Cantorian space-time, where the two symmetric times, 0 + it and 0 − it, coexist to our 4-dimensional smooth space-time, where t is the only time.     

El Naschie’s cantorian strings and duality in Weyl–Dirac theory

In the weak-field approximation, some implications of duality in the Weyl–Dirac (WD) theory, using the Gregorash–Papini–Wood approach, are investigated. Any particle is in a permanent interaction with the ‘subquantic level’ (Madelung’s fluid) and, as a result of this interaction, the particle acquires the proper fluctuation curvature and the proper fluctuation energy, respectively. By fixing the fluctuations scale, the quantum fluid orders either by means of bright cnoidal oscillation modes inducing causality, or by means of dark cnoidal oscillation modes inducing acausality, and non-linear effects, respectively. The periodic mode is associated with the undulatory characteristic, and the solitonic one with the corpuscular one. By not fixing the fluctuations scale and keeping the symmetry, the quantum fluid orders like a two-dimensional (2D) lattice of vortices, so that the duality needs coherence. In the compatibility between quantum hydrodynamics in the Madelung’s representation and the wave mechanics, the self-gravitational field of the Weyl–Dirac type physical object is generated. El Naschie’s space–time implies, by means of transfinite heterotic string theory, the masses of nucleons, and, by the gravitational fractional quantum Hall effect, the dispersion of the wave-packet on the particle. The analysis of the fractal dimension of the physical object described by the WD theory shows that the waves, and corpuscle, respectively are 2D projections of a higher dimensional special string in El Naschie’s space–time (El Naschie’s string).     

Derivation of the energy–momentum and Klein–Gordon equations from El Naschie’s complex time

In a previous note, we have provided a formal derivation of the transverse Doppler shift of special relativity from the generalization of El Naschie’s complex time. Here, we show that the relativistic energy–momentum equation, and hence the Klein–Gordon equation, are also natural consequences of the complex time generalization.     

El Naschie’s ε theory and effects of nanoparticle clustering on the heat transport in nanofluids

Effects of nanoparticle clustering on the heat transfer in nanofluids using the scale relativity theory in the topological dimension D_T = 3 are analyzed. In the one-dimensional differentiable case, the clustering morphogenesis process is achieved by cnoidal oscillation modes of the speed field. In such conjecture, a non-autonomous regime implies a relation between the radius and growth speed of the cluster while, a quasi-autonomous regime requires El Naschie’s ε^(∞) theory through the cluster–cluster coherence (El Naschie global coherence). Moreover, these two regimes are separated by the golden mean. In the one-dimensional non-differentiable case, the fractal kink spontaneously breaks the ‘vacuum symmetry’ of the fluid by tunneling and generates coherent structures. This mechanism is similar to the one of superconductivity. Thus, the fractal potential acts as an energy accumulator while, the fractal soliton, implies El Naschie’s ε^(∞) theory (El Naschie local coherence). Since all the properties of the speed field are transferred to the thermal one, for a certain conditions of an external load (e.g. for a certain value of thermal gradient) the soliton and fractal one breaks down (blows up) and release energy. As result, the thermal conductibility in nanofluids unexpectedly increases. Here, El Naschie’s ε^(∞) theory interferes through El Naschie global and local coherences.     

El Naschie’s ε space–time, hydrodynamic model of scale relativity theory and some applications

A generalization of the Nottale’s scale relativity theory is elaborated: the generalized Schrödinger equation results as an irrotational movement of Navier–Stokes type fluids having an imaginary viscosity coefficient. Then ψ simultaneously becomes wave-function and speed potential. In the hydrodynamic formulation of scale relativity theory, some implications in the gravitational morphogenesis of structures are analyzed: planetary motion quantizations, Saturn’s rings motion quantizations, redshift quantization in binary galaxies, global redshift quantization etc. The correspondence with El Naschie’s ε^(∞) space–time implies a special type of superconductivity (El Naschie’s superconductivity) and Cantorian-fractal sequences in the quantification of the Universe.     

An approximate Herbrand’s theorem and definable functions in metric structures

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite‐dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand's theorem can be used to characterize definable functions in absolutely ubiquitous structures from classical logic.     

Newton’s method’s basins of attraction revisited

In this paper, we revisit the chaotic number of iterations needed by Newton’s method to converge to a root. Here, we consider a simple modified Newton method depending on a parameter. It is demonstrated using polynomiography that even in the simple algorithm the presence and the position of the convergent regions, i.e. regions where the method converges nicely to a root, can be complicatedly a function of the parameter.     

Modified integral current methods in electrodynamics of nonhomogeneous media

The main difficulty in numerical solution of integral equations of electrodynamics is associated with the need to solve a high-order system of linear equations with a dense matrix. It is therefore relevant to develop numerical methods that lead to linear equation systems of lower order at the cost of more complex evaluation of the coefficients. In this article we propose a method for solving linear equations of electrodynamics which is a modification of the integral current method. The main distinctive feature of the proposed method is double integration of the electric Green’s tensor in the process of algebraization of the original integral equation. The solutions of the system of linear equations are thus integral means of the electric field inside the anomaly constructed by the proposed transformation formula. We prove convergence and derive error bounds for both the solution of the integral equation and the electromagnetic field components evaluated from approximate transformation formulas.     

Horseshoes in chromosome’s attractors

In this paper, we study dynamics of a class of chromosome’s attractors. We show that these chromosome sequences are chaotic by giving a rigorous verification for existence of horseshoes in these systems. We prove that the Poincaré maps derived from these chromosome’s attractors are semi-conjugate to the 2-shift map, and its entropy is no less than log 2. The chaotic behavior is robust in the following sense: chaos exists when one parameter varies from −5.5148 to −5.4988.     

Numerical solutions of the Laplace’s equation

This paper uses the sinc methods to construct a solution of the Laplace’s equation using two solutions of the heat equation. A numerical approximation is obtained with an exponential accuracy. We also present a reliable algorithm of Adomian decomposition method to construct a numerical solution of the Laplace’s equation in the form a rapidly convergence series and not at grid points. Numerical examples are given and comparisons are made to the sinc solution with the Adomian decomposition method. The comparison shows that the Adomian decomposition method is efficient and easy to use.     

Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge

Very often traditional approaches studying dynamics of self-similarity processes are not able to give their quantitative characteristics at infinity and, as a consequence, use limits to overcome this difficulty. For example, it is well known that the limit area of Sierpinski’s carpet and volume of Menger’s sponge are equal to zero. It is shown in this paper that recently introduced infinite and infinitesimal numbers allow us to use exact expressions instead of limits and to calculate exact infinitesimal values of areas and volumes at various points at infinity even if the chosen moment of the observation is infinitely faraway on the time axis from the starting point. It is interesting that traditional results that can be obtained without the usage of infinite and infinitesimal numbers can be produced just as finite approximations of the new ones. The importance of the possibility to have this kind of quantitative characteristics for E-Infinity theory is emphasized.     

Fan’s inequality in geodesic spaces

Fan’s minimax inequality is extended to the context of metric spaces with global nonpositive curvature. As a consequence, a much more general result on the existence of a Nash equilibrium is obtained.     

Lyapunov’s inequality on timescales

The purpose of this work is to establish the timescale version of Lyapunov’s inequality as follows: Let $x\left(t\right)$ be a nontrivial solution of ${\left(r\left(t\right)x^{\Delta }\left(t\right)\right)}^{\Delta }+p\left(t\right)x^{\sigma }\left(t\right)=0\text{on }[a,b]$ satisfying $x\left(a\right)=x\left(b\right)=0$. Then, under suitable conditions on $p$, $r$, $a$ and $b$, we have ${\int }_a^b{p}_{+}\left(t\right)\Delta t\ge \{\begin{array}{cc}\frac{r\left(a\right)}{r\left(b\right)}\frac{b-a}{f\left(d\right)}\text{,}& \text{if }r\text{ is increasing}\text{,}\\ \frac{r\left(b\right)}{r\left(a\right)}\frac{b-a}{f\left(d\right)}\text{,}& \text{if }r\text{ is decreasing}\text{,}\end{array}$ where $p_{+}\left(t\right)=max\{p\left(t\right),0\},f\left(t\right)=(t-a)(b-t)$ and $d\in \mathbb{T}$ satisfies $|\frac{a+b}{2}-d|=min\{|\frac{a+b}{2}-s|\mid s\in [a,b]\cap \mathbb{T}\}$ if $\frac{a+b}{2}\in \mathbb{T}$. Here $\mathbb{T}$ is a timescale (see below).     

Chaos synchronization of the fractional-order Chen’s system

In this paper, based on the stability theorem of linear fractional systems, a necessary condition is given to check the chaos synchronization of fractional systems with incommensurate order. Chaos synchronization is studied by utilizing the Pecora–Carroll (PC) method and the coupling method. The necessary condition can also be used as a tool to confirm results of a numerical simulation. Numerical simulation results show the effectiveness of the necessary condition.     

An optimization of Chebyshev’s method

From Chebyshev’s method, new third-order multipoint iterations are constructed with their efficiency close to that of Newton’s method and the same region of accessibility.     

Constrained versions of Sauer’s lemma

Let [n]={1,…,n}. For a function h:[n]→{0,1}, x[n] and y{0,1} define by the width ω_h(x,y) of h at x the largest nonnegative integer a such that h(z)=y on x−a≤z≤x+a. We consider finite VC-dimension classes of functions h constrained to have a width ω_h(x_i,y_i) which is larger than N for all points in a sample or a width no larger than N over the whole domain [n]. Extending Sauer’s lemma, a tight upper bound with closed-form estimates is obtained on the cardinality of several such classes.     

Generalizations of Heilbronn’s triangle problem

For given integers d,j≥2 and any positive integers n, distributions of n points in the d-dimensional unit cube [0,1]^d are investigated, where the minimum volume of the convex hull determined by j of these n points is large. In particular, for fixed integers d,k≥2 the existence of a configuration of n points in [0,1]^d is shown, such that, simultaneously for j=2,…,k, the volume of the convex hull of any j points among these n points is Ω(1/n^{(j−1)/(1+|d−j+1|)}). Moreover, a deterministic algorithm is given achieving this lower bound, provided that d+1≤j≤k.